Identifier
Identifier
Values
[] generating graphics... => 0
[1] generating graphics... => 0
[2] generating graphics... => 1
[1,1] generating graphics... => 0
[3] generating graphics... => 1
[2,1] generating graphics... => 1
[1,1,1] generating graphics... => 0
[4] generating graphics... => 1
[3,1] generating graphics... => 1
[2,2] generating graphics... => 1
[2,1,1] generating graphics... => 1
[1,1,1,1] generating graphics... => 0
[5] generating graphics... => 1
[4,1] generating graphics... => 1
[3,2] generating graphics... => 1
[3,1,1] generating graphics... => 1
[2,2,1] generating graphics... => 1
[2,1,1,1] generating graphics... => 1
[1,1,1,1,1] generating graphics... => 0
[6] generating graphics... => 1
[5,1] generating graphics... => 1
[4,2] generating graphics... => 2
[4,1,1] generating graphics... => 1
[3,3] generating graphics... => 1
[3,2,1] generating graphics... => 2
[3,1,1,1] generating graphics... => 1
[2,2,2] generating graphics... => 1
[2,2,1,1] generating graphics... => 1
[2,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1] generating graphics... => 0
[7] generating graphics... => 1
[6,1] generating graphics... => 1
[5,2] generating graphics... => 2
[5,1,1] generating graphics... => 1
[4,3] generating graphics... => 1
[4,2,1] generating graphics... => 2
[4,1,1,1] generating graphics... => 1
[3,3,1] generating graphics... => 1
[3,2,2] generating graphics... => 1
[3,2,1,1] generating graphics... => 2
[3,1,1,1,1] generating graphics... => 1
[2,2,2,1] generating graphics... => 1
[2,2,1,1,1] generating graphics... => 1
[2,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1] generating graphics... => 0
[8] generating graphics... => 1
[7,1] generating graphics... => 1
[6,2] generating graphics... => 2
[6,1,1] generating graphics... => 1
[5,3] generating graphics... => 2
[5,2,1] generating graphics... => 2
[5,1,1,1] generating graphics... => 1
[4,4] generating graphics... => 1
[4,3,1] generating graphics... => 1
[4,2,2] generating graphics... => 2
[4,2,1,1] generating graphics... => 2
[4,1,1,1,1] generating graphics... => 1
[3,3,2] generating graphics... => 1
[3,3,1,1] generating graphics... => 1
[3,2,2,1] generating graphics... => 2
[3,2,1,1,1] generating graphics... => 2
[3,1,1,1,1,1] generating graphics... => 1
[2,2,2,2] generating graphics... => 1
[2,2,2,1,1] generating graphics... => 1
[2,2,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1] generating graphics... => 0
[9] generating graphics... => 1
[8,1] generating graphics... => 1
[7,2] generating graphics... => 2
[7,1,1] generating graphics... => 1
[6,3] generating graphics... => 2
[6,2,1] generating graphics... => 2
[6,1,1,1] generating graphics... => 1
[5,4] generating graphics... => 1
[5,3,1] generating graphics... => 2
[5,2,2] generating graphics... => 2
[5,2,1,1] generating graphics... => 2
[5,1,1,1,1] generating graphics... => 1
[4,4,1] generating graphics... => 1
[4,3,2] generating graphics... => 2
[4,3,1,1] generating graphics... => 1
[4,2,2,1] generating graphics... => 2
[4,2,1,1,1] generating graphics... => 2
[4,1,1,1,1,1] generating graphics... => 1
[3,3,3] generating graphics... => 1
[3,3,2,1] generating graphics... => 2
[3,3,1,1,1] generating graphics... => 1
[3,2,2,2] generating graphics... => 1
[3,2,2,1,1] generating graphics... => 2
[3,2,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,1] generating graphics... => 1
[2,2,2,1,1,1] generating graphics... => 1
[2,2,1,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1] generating graphics... => 0
[10] generating graphics... => 1
[9,1] generating graphics... => 1
[8,2] generating graphics... => 2
[8,1,1] generating graphics... => 1
[7,3] generating graphics... => 2
[7,2,1] generating graphics... => 2
[7,1,1,1] generating graphics... => 1
[6,4] generating graphics... => 2
[6,3,1] generating graphics... => 2
[6,2,2] generating graphics... => 2
[6,2,1,1] generating graphics... => 2
[6,1,1,1,1] generating graphics... => 1
[5,5] generating graphics... => 1
[5,4,1] generating graphics... => 1
[5,3,2] generating graphics... => 2
[5,3,1,1] generating graphics... => 2
[5,2,2,1] generating graphics... => 2
[5,2,1,1,1] generating graphics... => 2
[5,1,1,1,1,1] generating graphics... => 1
[4,4,2] generating graphics... => 2
[4,4,1,1] generating graphics... => 1
[4,3,3] generating graphics... => 1
[4,3,2,1] generating graphics... => 3
[4,3,1,1,1] generating graphics... => 1
[4,2,2,2] generating graphics... => 2
[4,2,2,1,1] generating graphics... => 2
[4,2,1,1,1,1] generating graphics... => 2
[4,1,1,1,1,1,1] generating graphics... => 1
[3,3,3,1] generating graphics... => 1
[3,3,2,2] generating graphics... => 1
[3,3,2,1,1] generating graphics... => 2
[3,3,1,1,1,1] generating graphics... => 1
[3,2,2,2,1] generating graphics... => 2
[3,2,2,1,1,1] generating graphics... => 2
[3,2,1,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,2] generating graphics... => 1
[2,2,2,2,1,1] generating graphics... => 1
[2,2,2,1,1,1,1] generating graphics... => 1
[2,2,1,1,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[11] generating graphics... => 1
[10,1] generating graphics... => 1
[9,2] generating graphics... => 2
[9,1,1] generating graphics... => 1
[8,3] generating graphics... => 2
[8,2,1] generating graphics... => 2
[8,1,1,1] generating graphics... => 1
[7,4] generating graphics... => 2
[7,3,1] generating graphics... => 2
[7,2,2] generating graphics... => 2
[7,2,1,1] generating graphics... => 2
[7,1,1,1,1] generating graphics... => 1
[6,5] generating graphics... => 1
[6,4,1] generating graphics... => 2
[6,3,2] generating graphics... => 2
[6,3,1,1] generating graphics... => 2
[6,2,2,1] generating graphics... => 2
[6,2,1,1,1] generating graphics... => 2
[6,1,1,1,1,1] generating graphics... => 1
[5,5,1] generating graphics... => 1
[5,4,2] generating graphics... => 2
[5,4,1,1] generating graphics... => 1
[5,3,3] generating graphics... => 2
[5,3,2,1] generating graphics... => 3
[5,3,1,1,1] generating graphics... => 2
[5,2,2,2] generating graphics... => 2
[5,2,2,1,1] generating graphics... => 2
[5,2,1,1,1,1] generating graphics... => 2
[5,1,1,1,1,1,1] generating graphics... => 1
[4,4,3] generating graphics... => 1
[4,4,2,1] generating graphics... => 2
[4,4,1,1,1] generating graphics... => 1
[4,3,3,1] generating graphics... => 1
[4,3,2,2] generating graphics... => 2
[4,3,2,1,1] generating graphics... => 3
[4,3,1,1,1,1] generating graphics... => 1
[4,2,2,2,1] generating graphics... => 2
[4,2,2,1,1,1] generating graphics... => 2
[4,2,1,1,1,1,1] generating graphics... => 2
[4,1,1,1,1,1,1,1] generating graphics... => 1
[3,3,3,2] generating graphics... => 1
[3,3,3,1,1] generating graphics... => 1
[3,3,2,2,1] generating graphics... => 2
[3,3,2,1,1,1] generating graphics... => 2
[3,3,1,1,1,1,1] generating graphics... => 1
[3,2,2,2,2] generating graphics... => 1
[3,2,2,2,1,1] generating graphics... => 2
[3,2,2,1,1,1,1] generating graphics... => 2
[3,2,1,1,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,2,1] generating graphics... => 1
[2,2,2,2,1,1,1] generating graphics... => 1
[2,2,2,1,1,1,1,1] generating graphics... => 1
[2,2,1,1,1,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
[12] generating graphics... => 1
[11,1] generating graphics... => 1
[10,2] generating graphics... => 2
[10,1,1] generating graphics... => 1
[9,3] generating graphics... => 2
[9,2,1] generating graphics... => 2
[9,1,1,1] generating graphics... => 1
[8,4] generating graphics... => 2
[8,3,1] generating graphics... => 2
[8,2,2] generating graphics... => 2
[8,2,1,1] generating graphics... => 2
[8,1,1,1,1] generating graphics... => 1
[7,5] generating graphics... => 2
[7,4,1] generating graphics... => 2
[7,3,2] generating graphics... => 2
[7,3,1,1] generating graphics... => 2
[7,2,2,1] generating graphics... => 2
[7,2,1,1,1] generating graphics... => 2
[7,1,1,1,1,1] generating graphics... => 1
[6,6] generating graphics... => 1
[6,5,1] generating graphics... => 1
[6,4,2] generating graphics... => 3
[6,4,1,1] generating graphics... => 2
[6,3,3] generating graphics... => 2
[6,3,2,1] generating graphics... => 3
[6,3,1,1,1] generating graphics... => 2
[6,2,2,2] generating graphics... => 2
[6,2,2,1,1] generating graphics... => 2
[6,2,1,1,1,1] generating graphics... => 2
[6,1,1,1,1,1,1] generating graphics... => 1
[5,5,2] generating graphics... => 2
[5,5,1,1] generating graphics... => 1
[5,4,3] generating graphics... => 2
[5,4,2,1] generating graphics... => 2
[5,4,1,1,1] generating graphics... => 1
[5,3,3,1] generating graphics... => 2
[5,3,2,2] generating graphics... => 2
[5,3,2,1,1] generating graphics... => 3
[5,3,1,1,1,1] generating graphics... => 2
[5,2,2,2,1] generating graphics... => 2
[5,2,2,1,1,1] generating graphics... => 2
[5,2,1,1,1,1,1] generating graphics... => 2
[5,1,1,1,1,1,1,1] generating graphics... => 1
[4,4,4] generating graphics... => 1
[4,4,3,1] generating graphics... => 1
[4,4,2,2] generating graphics... => 2
[4,4,2,1,1] generating graphics... => 2
[4,4,1,1,1,1] generating graphics... => 1
[4,3,3,2] generating graphics... => 2
[4,3,3,1,1] generating graphics... => 1
[4,3,2,2,1] generating graphics... => 3
[4,3,2,1,1,1] generating graphics... => 3
[4,3,1,1,1,1,1] generating graphics... => 1
[4,2,2,2,2] generating graphics... => 2
[4,2,2,2,1,1] generating graphics... => 2
[4,2,2,1,1,1,1] generating graphics... => 2
[4,2,1,1,1,1,1,1] generating graphics... => 2
[4,1,1,1,1,1,1,1,1] generating graphics... => 1
[3,3,3,3] generating graphics... => 1
[3,3,3,2,1] generating graphics... => 2
[3,3,3,1,1,1] generating graphics... => 1
[3,3,2,2,2] generating graphics... => 1
[3,3,2,2,1,1] generating graphics... => 2
[3,3,2,1,1,1,1] generating graphics... => 2
[3,3,1,1,1,1,1,1] generating graphics... => 1
[3,2,2,2,2,1] generating graphics... => 2
[3,2,2,2,1,1,1] generating graphics... => 2
[3,2,2,1,1,1,1,1] generating graphics... => 2
[3,2,1,1,1,1,1,1,1] generating graphics... => 2
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[2,2,2,2,2,2] generating graphics... => 1
[2,2,2,2,2,1,1] generating graphics... => 1
[2,2,2,2,1,1,1,1] generating graphics... => 1
[2,2,2,1,1,1,1,1,1] generating graphics... => 1
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 1
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 1
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 0
click to show generating function       
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$ \frac{1}{2}(\sqrt{1+8n}-3) $$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
References
[1] Brylawski, T. The lattice of integer partitions MathSciNet:0325405
Code
@cached_function
def P(k):
    return posets.IntegerPartitionsDominanceOrder(k)

def statistic(pi):
    Q = P(pi.size())
    return len(Q.lower_covers(Q(pi)))

Created
May 09, 2016 at 09:22 by Martin Rubey
Updated
Oct 29, 2017 at 21:36 by Martin Rubey